Polynomial maps on the monoid of words

We briefly visit the theory of polynomial and semipolynomial maps defined on an arbitrary monoid, with range a commutative group. Then we characterize the space $\mathcal{P}(S,\mathbb{C})$ of polynomial maps $f:S\to \mathbb{C}$, where $S=\mathcal{A}^*$ is the monoid of words based on an arbitrary al